Proposition REF  is proved using ideas from additive combinatorics. We will give a brief sketch of the proof, which is inspired by the proof of the Elekes-Rónyai theorem in [1]}. The goal is to prove an estimate of the form \(\mathcal {E}_{\delta }(\lbrace P(a,b)=P(a^{\prime },b^{\prime })\rbrace )\le \delta ^{-3\alpha +\varepsilon }\)  for some \(\varepsilon =\varepsilon (\alpha )>0\) . Here and for the rest of this proof sketch, we will always suppose that \(a,a^{\prime }\in A\)  and \(b,b^{\prime }\in B\) . Suppose instead that \(\mathcal {E}_{\delta }(\lbrace P(a,b)=P(a^{\prime },b^{\prime })\rbrace )\ge \delta ^{-3\alpha +\varepsilon }\) ; if \(\varepsilon >0\)  is sufficiently small we will obtain a contradiction.
